Factoring is used in solving many equations. If the factoring seems difficult, refer to the factoring section.

 

 

 

Example

 

Solve 2x² + 8 x = 0

 

The first thing to do in this problem is to factor. Notice that both terms have 2x in common.

 

2 x (x + 4) = 0

 

This is the same problem only factored. To solve from here, look at the two parts, 2x and x + 4, if either one of them is 0 then this equation is true. In other words, if 2 x = 0 then it does not matter what (x + 4) is, and vice versa (this is because anything times 0 is 0).

 

2 x = 0

 

Solve this and find out that x = 0 but there is also the case of (x + 4) = 0.

x + 4 = 0

 

Solve this and see that x = -4.

 

Now there are two possibilities for x. x = 0 and x = -4. They are both right.

x = -4, 0

 

Answer!

Note- Always write the answers from smallest to largest.

 

 

 

 

 

 

 

 

Example

 

Solve x² +3 x –10= 0

 

The first thing to do is to factor. Realize that it is a trinomial with a =1 (that makes it the easy kind), b = 3, and c = -10. Remember, look for multiples of –10 that add up to 3. Those two numbers are 5,-2.

 

(x + 5) (x – 2) = 0

 

As explained in the previous problem, (x + 5) = 0 or (x – 2) = 0 would make this equation true.

Look at them one at a time.

(x + 5) = 0 therefore x = -5

(x – 2) = 0 therefore x = 2

x = -5, 2

 

Answer!

 

Note- Anytime when solving a problem like this, make sure that it is equal to 0, or it will not work this way. An example of that is shown next.

 

 

 

 

 

 

Example

 

Solve x² - 5 x + 3= -1

 

The first thing to notice about this problem is that it is not equal to 0. Set it equal to 0 by adding 1 to both sides.

x² - 5 x + 4 = 0

 

Now that it is equal to 0, factor it.

 

(x – 4) (x – 1) = 0

 

Solve for x in the first possibility of (x – 4) = 0.

x = 4

 

Solve for the other case of (x –1) = 0.

x = 1

x = 1, 4

 

Answer!

 

 

 

 

 

Example

 

Solve x² – 25 = 0

 

Realize that this factors easily because it is a difference of 2 squares.

(x – 5) (x + 5) = 0

 

Solve for (x- 5) = 0 and for (x + 5) = 0.

 

x = -5, 5

 

Anytime there is an answer like this where there is a positive and a negative number that is the same, Rewrite it as follows.

 

x = ± 5

 

Answer! It is read as “Plus or minus 5” or “Positive or negative 5”.

 

 

 

 

 

 

 

 

Example

 

Solve 9x² + 6 x + 8 = 0

 

To factor realize that a = 9 and not 1 so this is one of the harder types. The description on how to factor this type is shown in the factoring chapter.

 

(3x – 2) (3x + 4) = 0

 

Solve for the cases (3 x – 2) = 0 and (3x + 4) = 0.

x = ,

 

Answer!

 

Note- Most people make the mistake here on looking at (3x –2) (3x + 4) = 0 and assuming the answer to be –4 and 2. This is not true. Solve for 3 x –2 = 0 by adding 2 to both sides which will result in 3 x = 2 and then divide by 2 to get . The same is true in getting .